Adding mesh to previously rigged object
You can add geometry after rigging without any need to delete your previous job.
If you used the automatic weights option, probably you will have to manual weight the new vertices: as Mentalist said, assign to the fingertip bone a weight of 1.0, but also check if there is some weight assigned to any other bone: if so you get deformations and you should remove these assignements.
In edit mode, if you select a vertex you can read wich are all his weight assignements in the object properties panel (Hotkey N).